They are identical in the sense that they have identical DNA in their nuclei. During a stage of the cell cycle called the S phase, the DNA in the chromosomes is replicated by the cell to make two identical copies of the entire genome of the cell. These identical chromosomes are called sister chromatids and are attached to one another. During mitosis, the sister chromatids are separated and two new nuclei form around each separate set. The cell then undergoes the final stage of the cell cycle called cytokinesis, where the single cell actually splits itself into two new cells. The cytoplasm and organelles of the one cell are split between the two new cells. So while the structure of the two new daughter cells may not be exactly the same, the DNA sequences in the chromosomes in the nuclei of the two cells are identical.
Because mitosis is the division of a single cell, the two daughter cells contain the same sequence of DNA and will therefore display the same genetic traits.